Question

an irrational number between √5 and 5

Answer 1

Answer:

√5 is an Irrational number.

Step-by-step explanation:

A rational number is a number which can be expressed in the form p/q where p and q are integers and q is not equal to 0. 5 can expressed in the form 5/1. Hence it is Rational and surely not Irrational.

√5

Let us suppose that √5 is rational.

Then it can be said that:

$\sqrt{5}=\frac{x}{y}$

Let us say x and y have some common factor.. Let us divide x and y with that factor to give:

$\sqrt{5}=\frac{a}{b}$

where a and b are co-prime

Now squaring both sides in this equation:

$(\sqrt{5})^{2} = \frac{a^{2}}{b^{2}} \\5 = \frac{a^{2}}{b^{2}}$

$5b^{2} = a^{2}$

In this equation, we can see that $b^{2}$ is a factor of $a^{2}$.

By Fundamental Theorem of Arithmetic, b is also a factor of a. This is a contradiction to the fact that a and b are co-prime.

Hence $\sqrt{5}$ is an irrational number.